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Critical Behavior of Charged Dilaton Black Holes in Ads Space

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Critical Behavior of Charged Dilaton Black Holes in Ads Space PHYSICAL REVIEW D 102, 064021 (2020) Critical behavior of charged dilaton black holes in AdS space Amin Dehyadegari1 and Ahmad Sheykhi1,2,3,* 1Physics Department and Biruni Observatory, Shiraz University, Shiraz 71454, Iran 2Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), P.O. Box 55134-441, Maragha, Iran 3Max-Planck-Institute for Gravitational Physics (Albert-Einstein-Institute), 14476 Potsdam, Germany (Received 26 February 2020; accepted 21 August 2020; published 9 September 2020) We revisit critical behavior and phase structure of charged anti–deSitter (AdS) dilaton black holes for arbitrary values of dilaton coupling α, and realize several novel phase behavior of this system. We adopt the viewpoint that cosmological constant (pressure) is fixed and treat the charge of the black hole as a thermodynamical variable. We study critical behavior and phase structure by analyzing the phase diagrams in T − S and q–T planes. We numerically derive the critical point in terms of α and observe that for α ¼ 1 pffiffiffi and α ≥ 3, the system does not admit any critical point, while for 0 < α < 1, the critical quantities are not α significantly affected by . We find that unstable behavior of the Helmholtz free energy for q<qc exhibits a first order (discontinuous) phase transition between small and large black holes for 0 ≤ α < 1, where q pffiffiffi c 1 α 3 is the value of charge at the critical point. For < < and q>qc, however, a novel first order phase transition occurs between small and large black hole, which has not been observed in the previous studies on phase transition of charged AdS black holes. DOI: 10.1103/PhysRevD.102.064021 I. INTRODUCTION Recently, it has been shown that for charged AdS black hole a second and first order phase transitions occur Phase transition is certainly one of the most intriguing between small and large black holes in an extended phase and interesting phenomena in the thermodynamic descrip- space which resembles the liquid-gas phase transition in the tion of black holes which may shed light on the nature of usual van der Waals liquid-gas system [3,4]. In an extended quantum gravity. In particular, the investigations on the thermodynamic phase space, the cosmological constant black hole phase structure/transition in an asymptotically (AdS length) is considered as a thermodynamic pressure AdS spacetime have received considerable attentions in the which can vary, and its corresponding conjugate quantity is past years. This is mainly due to the remarkable duality the thermodynamic volume of the black hole. Taking into between gravity in an (n þ 1)-dimensional AdS spacetime account the variation of pressure in the first law, one and the conformal field theory living on the boundary of observes that the mass of AdS black hole is equivalent its n-dimensional spacetime, (AdS=CFT) correspondence. to the enthalpy [5]. In the recent years, thermodynamic Perhaps, one of the earliest studies in this direction was phase transitions in an extended phase space have been done by Hawking and Page [1], who disclosed that there is explored for various types of black holes in AdS space (see indeed a first order phase transition, latter named Hawking- Refs. [6–22] and references therein). Reference [23] per- Page phase transition, between the thermal radiation and formed a stability analysis of charged AdS dilaton black the stable large Schwarzschild black hole with spherical hole by considering the second derivative of mass with horizon in the background of AdS spacetime. Later, Witten respect to entropy. It was shown that the black holes are discovered [2] that this phase transition can be interpreted thermodynamically stable for small α, while for large α in the AdS=CFT duality as the confinement/deconfinement black hole becomes unstable [23]. The studies on phase phase transition of the strongly coupled gauge theory. transition in such black hole have been carried out from both thermal and dynamical point of view [24], where the *[email protected] cosmological constant appears as a thermodynamical variable. They found that for small dilaton coupling, Published by the American Physical Society under the terms of α ≈ 0.01, the system resembles the van der Waals fluid the Creative Commons Attribution 4.0 International license. behavior, while for α > 1, the P − v diagram of the system Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, deviates and new phenomena beyond the van der Waals and DOI. Open access publication funded by the Max Planck liquid-gas-like appears [24]. Recently, a novel phase Society. behavior represents a small/large black hole zeroth-order 2470-0010=2020=102(6)=064021(9) 064021-1 Published by the American Physical Society AMIN DEHYADEGARI and AHMAD SHEYKHI PHYS. REV. D 102, 064021 (2020) phase transition, in an extended phase space with varying such that black hole has different behaviors for 0 < α < pffiffiffi pffiffiffi cosmological constant, has been observed for charged 1= 3 and 1= 3 < α. Besides, we calculate the critical dilaton black holes, where the geometry of spacetime is point numericallyffiffiffi for various values of α in T-S plane. When not asymptotically AdS [25]. p α ¼ 1 and α ≥ 3, we cannot realize any critical point in the Another possible approach to study thermodynamic system which has not been reported in the previous inves- phase structure of black hole is to consider the variation tigations on phase transition of charged AdS black holes. of electric charge of the black hole, while the cosmological Besides, for 0 ≤ α < 1, we observe a small/large first constant (AdS length) is kept fixed. With regard to this order phase transition for q<q, while similar behavior perspective, the critical behavior and phase transition of pffiffiffi c is seen for 1 < α < 3 provided q>q. Indeed, there is a charged AdS black holes were investigated in a fixed AdS c pffiffiffi 1 α 3 geometry, indicating that it exhibits the small/large black novel first order phase transition in the range of < < hole phase transition of van der Waals type [26–28]. which has not previously been observed. Finally, the phase Interestingly enough, it has been realized [26] that the diagram for charged AdS dilaton black hole is constructed phase transition of charged AdS black hole can occur in in q-T plane. 2 − Ψ Ψ 1 2 2 This paper is structured in the following manner. In Q plane, where ¼ = rþ is the conjugate of Q , without extending the phase space. Indeed, in this alter- Sec. II, a brief overview on thermodynamics of the four- native perspective, the relevant response function clearly dimensional charged dilaton black hole in the AdS back- signifies the stable and unstable region. Also there still exist ground is given. In Sec. III, we investigate critical behavior of charged dilaton AdS black holes by studying the specific a deep analogy between critical phenomena and critical heat at constant electric charge in T-S plane. In Sec. IV,we exponents of the system with those of van der Waals liquid- use the Helmholtz free energy to determine the possible gas system [26]. The advantages of this new approach is phase transition in the system. Finally, we summarize the that one do not need to extend the phase space by treating main results Sec. V. the cosmological constant as a thermodynamical variable which may physically make no sense [26]. It has been confirmed that this new approach also works in other II. CHARGED DILATON BLACK HOLES gravity theories [29,30] as well as in higher spacetime IN AdS SPACE dimensions [31]. Recently, the universality class and We start with a brief review on charged AdS black holes critical properties of any AdS black hole, independent of in dilaton gravity and calculate the associated conserved spacetime metric, via an alternative phase space has been and thermodynamic quantities. Exact charged dilaton black explored [32]. It was shown that the values of critical hole solutions in the background of (A)dS space in four exponents for generic black hole are the same as ones of [35] and higher dimensions [36,37] have been presented the van der Waals fluid system [32]. For Born-Infeld AdS by Gao and Zhang. Thermodynamics of these solutions black holed in four dimensional spacetime, we analytically has been investigated in [23,38]. The four-dimensional calculated the critical point by studying the behavior of action of Einstein-Maxwell gravity coupled to a dilaton specific heat in a fixed AdS geometry [33]. Furthermore, field is [23,35] the interesting reentrant phase transition of Born-Infeld Z AdS black hole has been investigated in the thermodynamic 1 ffiffiffiffiffiffi S − 4 p− R − 2 ∇φ 2 − φ phase space [33]. The structure of charged black holes has ¼ 16π d x gð ð Þ Vð Þ been explored by employing the Ruppeiner geometry [34] −2αφ μν − e FμνF ; 1 in which the charge is allowed to vary. Þ ð Þ In this paper, we study thermodynamic properties of where R is the Ricci scalar curvature, φ is the dilaton field charged AdS dilaton black hole including the phase tran- and VðφÞ is the dilaton potential. Herein, the electromag- sition and critical behavior in (3 þ 1)-dimensional space- netic field tensor Fμν is defined in terms of the gauge field time. Our work differs from [24] in that we keep the ∂ − ∂ cosmological constant as a fixed quantity and treat the Aμ via Fμν ¼ μAν νAμ. For an arbitrary value of the α charge of the black hole as a thermodynamic variable, while dilaton coupling strength in AdS space, the dilaton the authors of [24] extended the phase space by treating potential is chosen to take the following form [35,39] cosmological constant as a variable.
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